Abstract

We consider the problem of extending functions phi : S-n --> S-n to functions u : Bn+1 --> S-n for n = 2, 3. We assume phi belongs to the critical space W-1,W-n and we construct a W-1,W-(n+1,W-infinity)-controlled extension u. The Lorentz-Sobolev space W (1,(n+1,infinity)) is optimal for such controlled extension. Then we use these results to construct global controlled gauges for L-4-connections over trivial SU(2)-bundles in 4 dimensions. This result is a global version of the local Sobolev control of connections obtained by K. Uhlenbeck.